CS2262 Spring 2007Assignment 6: Fixed Point Iteration and Interpolation1. To be completed before Tuesday 6th March Class: Work through Atkinson & Han Pages117 —128.2. (a) Calculate the first six iterates in the iterationxn+1= 1 + 0.3 sin(xn)with x0= 1.Choose other initial guesses x0and repeat this calculation.(b) Find an interval [a, b] satisfying the propertya ≤ x ≤ b implies that a ≤ g(x) ≤ bwhere g(x) is such that xn+1= g(xn).Hint: Leta = min[−∞<x<∞]g(x), b = max[−∞<x<∞]g(x)(c) Prepare a table of results from the fixed point iteration method with columns n, xn, α −xn,rn. Where n is the iteration number (n ≥ 0), α is the fixed point, and rnis t he ratio of theerror (rn= (α − xn)/(α − xn−1)).3. For the functions g(x) shown in Figures 23 and 24, draw on successive iterations of the fixed pointiteration method starting from an initial estimate x = x + 0.xyx0y=g(x)y=xFigure 23: Draw on iterations of the fixed point iteration method starting from x = x0xyx0y=g(x)Figure 24: Draw on iterations of the fixed point iteration method starting from x = x04. What are the solutions α, if any, of the equation x =√1 + x? Does the iteration xn+1=√1 + xnconverge to any of these solutions (assuming x0is chosen sufficiently close to α)?5. Consider the rootfinding problem f (x) = 0 with root α, with f#(x) %= 0. Convert it to thefixed-point problemx = x + cf(x) ≡ g(x)with c a nonzero constant. How should c be chosen to ensure rapid convergence ofxn+1= xn+ cf(xn)to α (provided that x0is chosen sufficiently close to α)? Apply your way of choosing c to therootfinding problem x3− 5 = 0.6. Use Aitken’s error estimation formula (3.53) to estimate the error α−x2in the following iterations:(a)xn+1= e−xn, x0= 0.57(b)xn+1=0.51 + x4n, x0= 0.48(c)xn+1= 1 + 0.5 sin(xn), x0= 1.57. For slowing convergent series, the Aitken extrapolation formula can greatly accelerate the conver-gence. Use the following algorithm:x1= g(x0)x2= g(x1)x3= Aitken extrapolate of x0, x1, and x2x4= g(x3)x5= g(x4)x6= Aitken extrapolate of x3, x4, and x5Continue this process in the same manner. Apply it to the following iterations(a)xn+1= 2e−xn, x0= 0.8(b)xn+1=0.91 + x4n, x0= 0.75(c)xn+1= 6.28 + sin(xn), x0= 6Due March 8th 2007Email completed assignments to